the equity premium puzzle bocong du november 18, 2013 chapter 13 ls 1/25
TRANSCRIPT
The Equity Premium Puzzle
Bocong Du
November 18, 2013 Chapter 13 LS 1/25
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
Framework:
• Prepare: Interpretation of risk-aversion parameter
• The equity premium puzzle ---- Issue raised
• Two statements of the equity premium puzzle
• A parametric statement
• A non-parametric statement
• The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 2/25
Interpretation of risk-aversion parameter
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 3/25
• CRRA Utility function:
• The individual’s coefficient of relative risk aversion:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 4/25
• Consider offering two alternative to a consumer who starts off with risk-free consumption level c:
Receive :• c-π with certainty
Receive: • c-y with probability 0.5• c+y with probability 0.5
• Aim: given y and c, we want to find the function π(y, c) that solves:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 5/25
• Taking the Taylor series expansion of LHS:
• Taking the Taylor series expansion of RHS:
• LHS=RHS:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 6/25
• In CRRA case, we get:
• Another form:
•
• Discussion of macroeconomists' prejudices about
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 7/25
The Equity Premium Puzzle
• • •
: The real return to stock
: The real return to relatively riskless bonds
: The growth rate of per capita real consumption of nondurables and services
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 8/25
A Parametric Statement of the Equity Premium Puzzle
• Starting from Euler Equations:
• Assumption:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 9/25
• Substituting CRRA and the stochastic processes into Euler Equation:
• Taking logarithms:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 10/25
• Taking the difference between the expressions for rs and rb:
• Approximation:
= 0 From Table 10.2 (-0.000193)
• Then we get:
0.06 0.00219
27.40 The Equity Premium Puzzle
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 11/25
A Non-Parametric Statement of the Equity Premium Puzzle
: Time-t price of the asset•
• : one-period payoff of the asset
• : stochastic discount factor for discounting the stochastic payoff (price kernel)
Market Price of Risk:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 12/25
• Apply Cauchy-Schwarz inequality:
Market Price of Risk•
•
: the reciprocal of the gross one-period risk-free return by setting
: a conditional standard deviation
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 13/25
Hansen-Jagannathan bounds:
• Construct structural models of the stochastic discount factor
• Construct x, c, p, q, and π
• Inner product representation of the pricing kernel
• Classes of stochastic discount factors
• A Hansen-Jagannathan bound: One example
• The Mehra-Prescott data ---- HJ statement of the equity premium puzzle
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 14/25
Construct structural models of the stochastic discount factor
• Construct x, c, p, q, and π
x=
X1
X2
X1
.
.
.XJ J×1
J basic securities
x: random vector of payoffs on the basic securities
C= C1 C2 C3 … CJ
1×Jc: a vector of portfolio weights
p = c · xp: portfolio
We seek a price functional q = π(x) qj = π(xj)
q: price of the basic securities
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 15/25
• The law of one price:
Which means the pricing functional π is linear on P
• Tow portfolios with the same payoff have the same price:
π(c, x) depends on c · x, not on c
• If x is return, then q=1, the unit vector, and:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 16/25
Construct structural models of the stochastic discount factor• Inner product representation of the pricing kernel
E(y·x) : the inner product of x and y x is the vector y is a scalar random variable
• Riesz Representation Theorem proves the existence of y in the linear functional
Definition: A stochastic discount factor is a scalar random variable y that satisfied the following equation:
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 17/25
• The vector of prices of the primitive securities, q, satisfies:
Where C= 1, 1, 1 … 11×J
• There exist many stochastic discount factors
• Classes of stochastic discount factors
Note:
The expected discount factor is the price of a sure scalar payoff of unity
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 18/25
Classes of stochastic discount factors
• Example 1:
• Example 2:
• Example 3:
• Example 4:
• A special case: Excess Returns
• A special case: q=1
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 19/25
A Hansen-Jagannathan bound: Example 4
• Given data on q and the distribution of returns x• A linear functional so y exits
e is orthogonal to x
• We know:
*
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 20/25
From:
Hansen-Jagannathan bound
Two specifications:
• For an excess return q = 0
• For a set of return q = 1
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 21/25
Excess Return: a return on a stock portfolio
: a return on a risk-free bond
So for an excess return, q = 0
*
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 22/25
Hansen-Jagannathan bound(This bound is a straight line)
When z is a scalar:
Market Price of Risk
• determines a straight-line frontier above which the stochastic discount factor must reside.
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
A Parametric StatementA Non-Parametric Statement Market Price of Risk Hansen-Jagannathan Bounds
November 18, 2013 Chapter 13 LS 23/25
For a set of return, q = 1
*
The Hansen-Jagannathan Bound(This bound is a parabola)
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 24/25
The Mehra-Prescott data
• The stochastic discount factor
• CRRA utility
• Data: annual gross real returns on stocks and bills in the United States for 1889 to 1979
FrameworkInterpretation of Risk-Aversion Parameter
The Equity Premium PuzzleTwo Statements
The Mehra-Prescott data
November 18, 2013 Chapter 13 LS 25/25
• Questions
• Comments